It sounds like the Accelerator pump ciruit is not working right..
The greater inertia of liquid gasoline, compared to air, means that if the throttle is suddenly opened, the airflow will increase more rapidly than the fuel flow, causing a temporary "lean" condition which causes the engine to "stumble" under acceleration (the opposite of what is normally intended when the throttle is opened). This is remedied by the use of a small mechanical pump, usually either a plunger or diaphragm type actuated by the throttle linkage, which propels a small amount of gasoline through a jet, wherefrom it is injected into the carburetor throat. This extra shot of fuel counteracts the transient lean condition on throttle tip-in. Most accelerator pumps are adjustable for volume and/or duration by some means. Eventually the seals around the moving parts of the pump wear such that pump output is reduced; this reduction of the accelerator pump shot causes stumbling under acceleration until the seals on the pump are renewed.
The accelerator pump is also used to prime the engine with fuel prior to a cold start. Excessive priming, like an improperly-adjusted choke, can cause flooding. This is when too much fuel and not enough air are present to support combustion. For this reason, some carburetors are equipped with an unloader mechanism: The accelerator is held at wide open throttle while the engine is cranked, the unloader holds the choke open and admits extra air, and eventually the excess fuel is cleared out and the engine starts.
